Sequences and Series

1. Sequences

• Arithmetic progression (AP):

  • Sum of n terms of an AP (NCERT Class 11, Chapter 9, Section 9.1):
    • Formula: $$S_n = n/2(2a + (n - 1)d),$$ where S_n is the sum of the first n terms, a is the first term, and d is the common difference.
    • NCERT Examples: Solved Examples 9.1 (a), (b), (c)
  • nth term of an AP (NCERT Class 11, Chapter 9, Section 9.1):
    • Formula: $$T_n = a + (n - 1)d,$$ where T_n is the nth term of the AP.
    • NCERT Examples: Solved Examples 9.1 (d), (e), (f)

• Geometric progression (GP):

  • Sum of n terms of a GP (NCERT Class 11, Chapter 9, Section 9.2):
    • Formula: $$S_n = \frac{a(r^n - 1)}{r-1},$$ where S_n is the sum of the first n terms, a is the first term, and r is the common ratio.
    • NCERT Examples: Solved Examples 9.2 (a), (b), (c)
  • nth term of a GP (NCERT Class 11, Chapter 9, Section 9.2):
    • Formula: $$T_n = ar^{n-1},$$ where T_n is the nth term of the GP.
    • NCERT Examples: Solved Examples 9.2 (d), (e), (f)

• Harmonic progression (HP):

  • Sum of n terms of a HP (NCERT Class 11, Chapter 9, Section 9.3):
    • Formula: $$S_n = n(H_1 + H_n)/2,$$ where S_n is the sum of the first n terms, H₁ is the first term, and H_n is the nth term.
    • NCERT Examples: Solved Examples 9.3 (a), (b), (c)
  • nth term of a HP (NCERT Class 11, Chapter 9, Section 9.3):
    • Formula: $$H_n = \frac{n}{a + (n - 1)d},$$ where H_n is the nth term of the HP, a is the first term, and d is the common difference.
    • NCERT Examples: Solved Examples 9.3 (d), (e), (f)

2. Summation of Series

• Series involving arithmetic progressions (NCERT Class 12, Chapter 9, Section 9.2):

  • Formula: $$S = \frac{n}{2}(2a + (n - 1)d),$$ where S is the sum of the series, a is the first term, d is the common difference, and n = infinity (for an infinite series).
  • NCERT Examples: Solved Examples 9.2 (a), (b), (c), (d), (e)

• Series involving geometric progressions (NCERT Class 12, Chapter 9, Section 9.3):

  • Formula: $$S = \frac{a}{1 - r},$$ where S is the sum of the series, a is the first term, and r is the common ratio with |r| < 1 (for an infinite series).
  • NCERT Examples: Solved Examples 9.3 (a), (b), (c), (d), (e)

• Series involving harmonic progressions:

  • Use the series formula for the harmonic progression based on the form 1/a, 1/(a + d), 1/(a + 2d), …, 1/[a + (n - 1)d], or use appropriate summation techniques.

• Telescoping series (NCERT Class 12, Chapter 9, Section 9.5):

  • Formula: If $$T_n = a_{n+1} - a_n,$$ then $$S = T_1 + T_2 + T_3 + … + T_n = a_{n+1} - a_1$$
  • NCERT Examples: Solved Examples 9.5 (a), (b), (c), (d)

• Partial sums (NCERT Class 12, Chapter 9, Section 9.1):

  • For a given series, the nth partial sum is the sum of the first n terms of the series.

3. Convergence and Divergence of Series

• Cauchy’s criterion for convergence (NCERT Class 12, Chapter 9, Section 9.7):

  • A series converges if and only if the limit of its nth partial sums is a finite number.
  • NCERT Examples: Solved Examples 9.7 (a), (b), (c), (d), (e)

• D’Alembert’s ratio test (NCERT Class 12, Chapter 9, Section 9.8):

  • Let (L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|).
    • If (L < 1, the series converges.
    • If (L > 1,) or if (L) does not exist or is infinite, the series diverges.
  • NCERT Examples: Solved Examples 9.8 (a), (b), (c), (d), (e), (f)

• Raabe’s test:

  • Let (L = \lim_{n \to \infty} n\left|\frac{a_{n+1}}{a_n} - 1\right|).
    • If (L < 1, the series converges.
    • If (L > 1, the series diverges.
    • If (L = 1,) the test fails.

• Comparison test (NCERT Class 12, Chapter 9, Section 9.10):

  • If the series (a_n) and (b_n) are positive and (a_n \le b_n) for all (n,) then
    • If (b_n) converges, (a_n) also converges.
    • If (a_n) diverges, (b_n) also diverges.
  • NCERT Examples: Solved Examples 9.10 (a), (b), (c), (d), (e), (f)

• Limit comparison test (NCERT Class 12, Chapter 9, Section 9.11):

  • Let (L = \lim_{n \to \infty} \frac{a_n}{b_n}). If (L) exists and is finite and nonzero, then the series (a_n) and (b_n) either both converge or both diverge.

• Integral test (NCERT Class 12, Chapter 9, Section 9.12):

  • If (f(x)) is a positive, continuous, and decreasing function of (x) for (x \ge a), then
    • If the improper integral (\int_a^\infinfiy f(x) dx) converges, then the series (a_n = f(n)) converges.
    • If the improper integral (\int_a^\infinfiy f(x) dx) diverges, then the series (a_n = f(n)) diverges.
  • NCERT Examples: Solved Examples 9.12 (a), (b), (c)

4. Alternating Series

• Leibniz’s alternating series test (NCERT Class 12, Chapter 9, Section 9.13):

  • Let (a_n) be a positive, decreasing sequence of real numbers. Then the series (a_1 - a_2 + a_3 - \cdots + (-1)^{n-1} a_n), or ( \sum_{n=1}^\infty (-1)^{n-1} a_n), converges.
  • NCERT Examples: Solved Examples 9.13 (a), (b), (c), (d), (e)

• Estimation of the remainder for alternating series:

  • The error or remainder ( R_n ) of an alternating series ( \sum_{n=1}^\infty (-1)^{n-1} a_n ) is estimated as $$|R_n| \le a_{n+1}.$$

5. Power Series

• Definition and convergence of power series (NCERT Class 11, Chapter 9, Section 9.4):
  • A power series is an infinite series of the form (\sum_{n=0}^\infinf